ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

DIVISION OF FRACTIONS 177 duct of the denominators is the (result of) multiplica- tion of two or more fractions.¹ While all other writers give the rule in the same way as Brahmagupta, Mahāvīra in the Gaṇitasārasaṃgraha refers to cross reduction in order to shorten the work : In the multiplication of fractions, the numerators are to be multiplied by the numerators and the denomi- nators by denominators, after carrying out the pro- cess of cross reduction, if that be possible.² Division of Fractions The Āryabhaṭīya does not explicitly give the rule of divi- sion, but under the Rule of Three, we have an indication of this operation. The Rule of Three states the result as (f × i) / p, where f stands for phala i.e. "fruit", i for icchā, i.e., demand or requisition, and p for pramāṇa i.e. argument. When these quantities are fractional, we get an expression of the form (a/b × c/d) / (m/n) for the evaluation of which Āryabhaṭa I states : The multipliers and the divisor are multiplied by the denominators of each other. These quantities are written in the following way ┌───────┐ │ a m │ │ b n │ ├───────┤ │ c │ │ d │ └───────┘ Transferring the denominators we have ┌───────┐ │ a m │ │ n b │ │ c d │ └───────┘ Performing multiplication, the result is anc / mbd. The above interpretation of the obscure line in the Āryabhaṭīya is based

1. रूपाणिच्छेद गुण्यान्यंशयुतानि द्वयोर्बहूनां वा । प्रत्युत्पन्नो भवति च्छेदवधेनोद्धृतोऽशवधः ॥ —BrSpSi. XII. 3
2. GSS. p. 25. (2)